From https://blog.justinmallone.com/sicp-12-procedures-and-the-processes-they-generate-part-1/
What exactly was the relevance of the golden ratio in understanding the inefficiency of the recursive fibonacci procedure? Quote of relevant material:
In fact, it is not hard to show that the number of times the procedure will compute
(fib 1)or(fib 0)(the number of leaves in the above tree, in general) is precisely Fib(n + 1). To get an idea of how bad this is, one can show that the value of Fib(n) grows exponentially with n . More precisely (see exercise 1.13), Fib(n) is the closest integer to \phi^5/\sqrt{n}, where …
From the book (for context):
Exercise 1.13. Prove that Fib(n) is the closest integer to \phi^n/\sqrt{5}, where \phi = (1+\sqrt{5})/2. Hint: let \psi = (1 - \sqrt{5})/2 Use induction and the definition of the Fibonacci numbers (see section 1.2.2) to prove that Fib(n) = (\phi^n - \psi^n)/\sqrt{5}.
I think the point of saying that Fib(n) \approx \phi^n/\sqrt{5} was to show specifically how Fib(n) grows exponentially with n and to ~foreshadow exercise 1.13.
Exercise 1.13 doesn’t seem that useful as a coding exercise, except insofar as rigorously analyzing the complexity of the fib function, via Fib. That skill seems more relevant for like academics and ppl doing theoretical comp sci. That sort of thing doesn’t come up in day-to-day programming (though being able to analyze an algorithms complexity does; in this case the complexity is O(c^n) for some constant c; edit: you don’t need to know what this means; I included it so that you could see the difference between more simplistic day-to-day complexity stuff and like in-depth analysis)
IDK if that answers your q